Stochastic Modeling of Coercivity - A Measure of Non-equilibrium State

a r X i v :c o n d -m a t /0507640v 1 [c o n d -m a t .s t a t -m e c h ] 27 J u l 2005

“Stochastic Modeling of Coercivity ”-A Measure of Non-equilibrium State.

S.Chakraverty and M.Bandyopadhyay

Nano Science Unit,S.N.Bose National Center for Basic Sciences,

JD Block,Sector III,Salt Lake City,Kolkata 700098,India

(Dated:February 2,2008)

A typical coercivity versus particle size curve for magnetic nanoparticles has been explained by using the Gilbert equation followed by the corresponding Fokker Plank equation.Kramer’s treatment has been employed to explain the increase in coercivity in the single domain region.The single to multi-domain transformation has been assumed to explain the decrease in coercive ?eld beyond a certain particle size.The justi?cation for using Langevin theory of paramagnetism (including anisotropy energy)to ?t the M vs H curve is discussed.The super-symmetric Hamiltonian approach is used to ?nd out the relaxation time for the spins (making an angle greater than 900with applied ?eld)at domain wall.The main advantage of our technique is that we can easily take into account the time of measurement as we usually do in realistic measurement.

PACS numbers:75.60.Ej,75.75.+a,75.60.Lr.

I.INTRODUCTION

Thermal excitation and relaxation phenomena play a very crucial role in the case of nanoparticles.At ?nite temperature,single domain nanoparticles often exhibit superparamagnetic behavior,i.e.,the relaxation time of the particles is much smaller than the characteristic time scale of the measuring instrument.Superparamagnetic behavior has currently been studied by a number of experimental techniques such as ac and dc susceptibility measurements [1,2,3,4],neutron di?raction [5]etc.These e?ects have considerable technological interest because of their relevance to the stability of information stored in the form of magnetized particles.

Coercivity is an important quantity which plays a crucial role as far as the stabilization of a magnetic system is concerned.Understanding the nature of the coercive ?eld with the variation of particle size is one of the central issues,which has been addressed in the present paper,by using stochastic theory [6].The behavior of the coercivity as a function of particle size is a old one[7].The physical phenomena creating the maximum in coercivity of nanoparticles as a function of size is a well known problem in magnetism.This problem has been qualitatively understood.Various theoretical models have been published on the particle size dependence of coercivity [8,9].Thermal switching in single-domain particles (where coercivity increases with size of particles)was considered by a lot of authors [10,11,12,13,14].These models failed to explain the de-crease in H c with the increase in V at large particle size.Nucleation of domain walls was investigated by Braun [15,16].The crossover from single to multi-domain switching was investigated numerically by Hinzke et al [17].The e?ect of measurement time (i.e.the time lag between the measurement and the application of ?eld)was not included in their approaches.In this paper we have emphasized to give a mathematical basis to explain this well known phenomena including the e?ect of measurement time.We quantitatively explain the

FIG.1:Coercive force as a function of particle size.(W.H.Meiklejohn,Rev.Mod.Phys.25,302(1953);F.E.Luborsky,J.Appl.Phys.32,171S(1961).)

peak in the coercivity vs.particle size curve.

In this paper we have tried to explain the non-monotonic (?rst increase and then decrease in coercivity,FIG.1)behavior in coercivity against the particle size with the help of non-equilibrium statistical mechanics approach.The time of measurement is automatically included in this approach.Our description is based on the assumption that our system is consist of mono-dispersed particles with no interparticle interaction.Although the particle size distribution and interparticle interaction can produce many interesting e?ects [19,20,21],but it is beyond the scope of this paper.

With the preceding background the paper is organized as follows.In Sec.II we have discussed the increasing part of the H c by assuming the material consist of single domain particle with high anisotropy barrier limit.This section also contains the magnetization calculation taking into account the anisotropy energy.In Sec.III we have discussed the decreasing part of the coercivity by using super symmetric quantum mechanics (SUSYQM)approach.Finally,in Sec.V we present our main conclusions about the signi?cance of the reported

2

FIG.2:Direction of the applied magnetic ?eld,magnetic easy axis and magnetization.

results.

II.SINGLE DOMAIN REGIME

The subject of how a bulk magnetic specimen acquires a single domain structure and exhibits magnetic viscos-ity due

to Neel

relaxation,when its size is reduced,is an old one [18].The critical volume of a single domain par-ticle with uniaxial high anisotropy energy as estimated by Kittel [22]is R c =(9σw /4πI s 2),where σw is the do-main wall energy per unit area and I s is the saturation magnetization of the particle.The numerical value of R c comes out to be ～10nm for K ～105erg/cc (K is the anisotropy energy per unit volume).Therefore the initial increase in coercivity is observed in single domain particles.This leads us to map our problem to that of an ensemble of uniaxial single domain particles in con-tact with a heat bath at temperature T.Let us assume that the magnetic moment vector of each particle ( μp )(FIG.2)makes an angle θm with the easy axis and this easy axis of the particle is at an angle θk with respect

to the applied magnetic ?eld( H

app ).The total energy of the system is given by:

E T =KV sin 2(θm )?μp H app cos (θk ?θm ).

(1)

The Gilbert equation [23]governing the dynamics of the spin is

d μp

? μp

?η

d M

k B T

.(3)E n e r g y

m

FIG.3:Energy pro?le of an uniaxial single domain mag-netic particle (a)in absence and (b)in presence of external magnetic ?eld.

Although we can solve the corresponding Fokker Plank equation with the help of SUSYQM.approach,we have used an approximate method to get a better feeling of the underlying physics of the problem.The number of particles oriented between θi ?θm to θi +θm is given by

n i =P (θi ,t )e

?

E T (θi )

k B T

sin (θm )dθm ,(5)

with i =1or 2.Since in our case the potential barrier (δE )is much higher than the thermal ?uctuation,we have used the saddle point approximation in Eq.(5)which results in

I i =

e

?

E T (θi )

C i

,(6)

where C i is the curvature at the i-th minima.Now the continuity equation for P (θm ,t )is given by

dP (θm ,t )

dt

.We have

also assumed the di?usion coe?cient to be space inde-pendent.In the limit of high relaxation time J ′(θm ,t )becomes independent of θand is given by

J qs (t )=?

sin (θm )

k B T

?E T (θm )

?θm

.

(9)

3

FIG.4:Computer simulated magnetization curve of two dif-

ferent particles(a)7nm(b)8.5nm,using Eq.(11).

Using Kramer’s ansatz in the above equation one can

write

dn2

τ2?

n1

k B T

τ1

+1

τ1+τ2

,(11)

where(n1?n2)is proportional to the net magnetiza-

tion along the direction of the applied magnetic?eld.

Eq.(11)needs one initial condition i.e.n2(0)to get the

value of n2(t).For a single domain particle with large

relaxation time,if one changes the magnetic?eld after

a?nite interval of time(t),then limδH→0?n H?δH

2(t)=

n H→2(0)=limδH→0+n H+δH

2

(t)=n H←

2

(0).This implies

that for a particular value of H one should not expect

to get the same value of magnetization during increas-ing and decreasing cycle of H.Since the relaxation time

τi increases with particle volume,M H←?M H→also in-creases with particle volume giving rise to higher coerciv-

ity.Hence coercivity is a consequence of the quasi-static non-equilibrium measurement.Therefore,Langevin the-ory of paramagnetism is not applicable in these cases. We have used Eq.(11)to numerically generate M vs H curve as shown in FIG.4,for particle size7nm and8.5 nm.We have used t=50sec and K=105erg/cc in the nu-merical calculation,which is realistic for measurements of coercivity by vibrating sample magnetometer.

In case of a superparamagnetic system,a common prac-tice is to?t the magnetization curve by using Langevin theory of paramagnetism[25].But it is not quite obvious, since Langevin theory of paramagnetism does not include the magnetocrystaline anisotropy energy.Therefore an attempt has been made here to calculate the magneti-zation of a superparamagnetic sample at equilibriums. Magnetization

=μp,(12) where,= 0πP(θ,H app)cosθsinθdθ,

P(θ,H app)=e

?E T(θ,H)

Z

,

E T=?Hμp cosθ+k sin2θ,

and partition function

Z= k1e?(k1+α22(Erfi[√2k+1)]

?Erfi[√2k?1)]),

(13) where k=KV(V is the volume of the particle),α= H appμp

k B T

.Here Er?is the error function [26].The magnetization of the system

=μp S[k1,α]?

α

2k1

+1)2?e k1(α

X

,(15) where X=

√k

1

(αk1(α

τ1

)

e?E1

e?E1k B T

,(16) where E1=E mag=?E2andτi=τ0e E ani?E i

k B T?e?E mag

e E mag k B T

.(17)

L

t b

a

FIG.5:Spin distribution near 1800domain wall (a)at real space (b)at probability space.

This indicates that the anisotropy energy does not a?ect the magnetization of superparamagnetic particle (with large anisotropy energy)in equilibrium.

III.MULTI-DOMAIN REGIME -SUPERSYMETRIC QUANTUM MECHANICS

(SUSYQM)APPROACH

Let us now concentrate on the second region where the coercivity ?eld decreases with the increase in parti-cle size.It is clear from the above discussion that this can not happen if the particles still comprise single do-mains.Since the coercivity of the single domain particle increases monotonically with the increase in particle vol-ume,hence a single to multi domain transformation takes place at the maximum of coercivity.

To explain the decreasing part of the coercivity ?eld let us consider an arrangement of spin in a linear chain as shown in FIG.5.In the following discussion one should keep in mind that we are not interested here in the origin of the domain wall,but we assume the existence of do-main and the Hamiltonian contains the relevant terms.The Gilbert equation corresponding to i-th spin is

d μi

? μi

?η

d μi

dt

=?h ′

k B T

?

k B T

?V (θ)

?θ

,

(19)

with h ′=?

γ02ημi

dz

=0,which physically means that there is no

spin hopping between two sites.This implies that the spin will start relaxing along the surface of the cylinder without changing its position along Z-axis.To solve the

above equation let us perform the following transforma-tion P (θ,t )=

?and ?=k B T .Using the transformation in Eq.(19)we have

dψ

2

?V

′2

2

?

V ′2

h 1

de?ning two operator A =

?

2?

and

A ?=??2?,such that ?A ?Aφ=λφwith ground state eigenvalue equal to zero (since Aφ0=0to get equilibrium distribution).It can be shown that if φ1be the ?rst excited eigenstate of A ?A then it is the ground state of AA ?with ground state eigenvalue λ1[27].Now one can apply variational method to get λ1

λ1=

φ1(θ)?AA ?φ1(θ)dθ

2Kθ

2

.Which gives us the

relaxation time of the order of h 1(??K ).The spins at the end of the domain wall have a double well like potential.For such double well potential we have λ1～h 1(e ?(V 0?V (θ1))+e ?(V 0?V (θ2)),where V 0is the barrier height and θ1and θ2are the position of the two minima.Now it is clear that the relaxation time depends on the damping parameter as well as the barrier height,which in turn depends on the value of anisotropy constant and the angle between successive spins.The anisotropy constant is higher (one order of magnitude)for smaller particle.So for a smaller particle it takes more time to relax back to its initial con?guration,giving rise to a higher coercive ?eld.The above model indicates that our system under consideration,consist of a linear chain of ferromagnetic particles having two domains,with their easy axes parallel to each other and with applied magnetic ?eld also.The above model also does not contain the domain of closure.Still the model can be regarded as the starting point to explain qualitatively the hysteresis of a multi domain system.

IV.CONCLUSION

The particle size dependence of the coercive ?eld is explained from the view point of non-equilibrium statis-tical mechanics.The zero hysteresis loss indicates that the system reaches its equilibrium state before the next data is taken.It is shown that the increase in coercivity is the e?ect of increase in relaxation time.The numer-ical calculation of hysteresis curve shows that the loss

as well as the coercive?eld increases with particle size. During the numerical calculation assumption has been made that k B T<δE.The Langevin theory of paramag-netism for superparamagnetic particle has been reestab-lished,after taking into account the magnetocrystaline anisotropy e?ect.Assuming the single to multi-domain transformation we have shown that the relaxation time of the samples decreases with increases in particle size due to decrease in surface pressure and anisotropy con-stant which gives a decrease in coercivity.The important point to note that in case of any experimental study of a single domain particle,one should perform this coercive ?eld(H c)vs.particle size curve and?gure out the peak in the curve.One should perform all the measurements below this peak particle size to analyze the behavior of single domain magnetic nanoparticles.

Acknowledgments

We wishes to express gratitude to Professor Sushanta Dattagupta for his illuminating course in non-equilibriums statistical mechanics and to Professor Binayak Dutta Ray for helping us to understand the underlying physics.We also grateful to Professor H.S. Mani for the constant encouragement.Financial sup-port from the Department of Science and Technology, Government of India is gratefully acknowledged.

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